Cofiniteness and non-vanishing of local cohomology modules

Abstract

Let $R$ be a commutative Noetherian local ring, $I$ an ideal of $R$, and let $M$ be a non-zero finitely generated $R$-module. In this paper, we establish some new properties of the local cohomology modules $H^i_I(M)$, $i\geq 0$. In particular, we show that if $(R,\mathfrak{m})$ is a Noetherian local integral domain of dimension $d \leq 4$ which is a homomorphic image of a Cohen-Macaulay ring and $x_1,\ldots,x_n$ is a part of a system of parameters for $R$, then for all $i\geq0$, the $R$-modules $H^i_{I}(R)$ are $I$-cofinite, where $I=(x_1,\ldots,x_n)$. Also, we prove that if $(R,\mathfrak{m})$ is a Noetherian local ring of dimension~$d$ and $x_1,\ldots,x_t$ is a part of a system of parameters for $R$, then $H^{d-t}_{\mathfrak{m}}(H^t_{(x_1,\ldots,x_t)}(R))\neq 0$. In particular, $\mu^{d-t}(\mathfrak{m},H^t_{(x_1,\ldots,x_t)}(R))\neq 0$ and ${\rm injdim}_R(H^t_{(x_1,\ldots,x_t)}(R))\geq d-t$.